1 /*- 2 * Copyright (c) 2012 Stephen Montgomery-Smith <stephen@FreeBSD.ORG> 3 * All rights reserved. 4 * 5 * Redistribution and use in source and binary forms, with or without 6 * modification, are permitted provided that the following conditions 7 * are met: 8 * 1. Redistributions of source code must retain the above copyright 9 * notice, this list of conditions and the following disclaimer. 10 * 2. Redistributions in binary form must reproduce the above copyright 11 * notice, this list of conditions and the following disclaimer in the 12 * documentation and/or other materials provided with the distribution. 13 * 14 * THIS SOFTWARE IS PROVIDED BY THE AUTHOR AND CONTRIBUTORS ``AS IS'' AND 15 * ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE 16 * IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE 17 * ARE DISCLAIMED. IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE 18 * FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL 19 * DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS 20 * OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) 21 * HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT 22 * LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY 23 * OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF 24 * SUCH DAMAGE. 25 */ 26 27 /* 28 * The algorithm is very close to that in "Implementing the complex arcsine 29 * and arccosine functions using exception handling" by T. E. Hull, Thomas F. 30 * Fairgrieve, and Ping Tak Peter Tang, published in ACM Transactions on 31 * Mathematical Software, Volume 23 Issue 3, 1997, Pages 299-335, 32 * http://dl.acm.org/citation.cfm?id=275324. 33 * 34 * See catrig.c for complete comments. 35 * 36 * XXX comments were removed automatically, and even short ones on the right 37 * of statements were removed (all of them), contrary to normal style. Only 38 * a few comments on the right of declarations remain. 39 */ 40 41 #include <sys/cdefs.h> 42 __FBSDID("$FreeBSD$"); 43 44 #include <complex.h> 45 #include <float.h> 46 47 #include "math.h" 48 #include "math_private.h" 49 50 #undef isinf 51 #define isinf(x) (fabsf(x) == INFINITY) 52 #undef isnan 53 #define isnan(x) ((x) != (x)) 54 #define raise_inexact() do { volatile float junk = 1 + tiny; } while(0) 55 #undef signbit 56 #define signbit(x) (__builtin_signbitf(x)) 57 58 static const float 59 A_crossover = 10, 60 B_crossover = 0.6417, 61 FOUR_SQRT_MIN = 0x1p-61, 62 QUARTER_SQRT_MAX = 0x1p61, 63 m_e = 2.7182818285e0, /* 0xadf854.0p-22 */ 64 m_ln2 = 6.9314718056e-1, /* 0xb17218.0p-24 */ 65 pio2_hi = 1.5707962513e0, /* 0xc90fda.0p-23 */ 66 RECIP_EPSILON = 1 / FLT_EPSILON, 67 SQRT_3_EPSILON = 5.9801995673e-4, /* 0x9cc471.0p-34 */ 68 SQRT_6_EPSILON = 8.4572793338e-4, /* 0xddb3d7.0p-34 */ 69 SQRT_MIN = 0x1p-63; 70 71 static const volatile float 72 pio2_lo = 7.5497899549e-8, /* 0xa22169.0p-47 */ 73 tiny = 0x1p-100; 74 75 static float complex clog_for_large_values(float complex z); 76 77 static inline float 78 f(float a, float b, float hypot_a_b) 79 { 80 if (b < 0) 81 return ((hypot_a_b - b) / 2); 82 if (b == 0) 83 return (a / 2); 84 return (a * a / (hypot_a_b + b) / 2); 85 } 86 87 static inline void 88 do_hard_work(float x, float y, float *rx, int *B_is_usable, float *B, 89 float *sqrt_A2my2, float *new_y) 90 { 91 float R, S, A; 92 float Am1, Amy; 93 94 R = hypotf(x, y + 1); 95 S = hypotf(x, y - 1); 96 97 A = (R + S) / 2; 98 if (A < 1) 99 A = 1; 100 101 if (A < A_crossover) { 102 if (y == 1 && x < FLT_EPSILON * FLT_EPSILON / 128) { 103 *rx = sqrtf(x); 104 } else if (x >= FLT_EPSILON * fabsf(y - 1)) { 105 Am1 = f(x, 1 + y, R) + f(x, 1 - y, S); 106 *rx = log1pf(Am1 + sqrtf(Am1 * (A + 1))); 107 } else if (y < 1) { 108 *rx = x / sqrtf((1 - y) * (1 + y)); 109 } else { 110 *rx = log1pf((y - 1) + sqrtf((y - 1) * (y + 1))); 111 } 112 } else { 113 *rx = logf(A + sqrtf(A * A - 1)); 114 } 115 116 *new_y = y; 117 118 if (y < FOUR_SQRT_MIN) { 119 *B_is_usable = 0; 120 *sqrt_A2my2 = A * (2 / FLT_EPSILON); 121 *new_y = y * (2 / FLT_EPSILON); 122 return; 123 } 124 125 *B = y / A; 126 *B_is_usable = 1; 127 128 if (*B > B_crossover) { 129 *B_is_usable = 0; 130 if (y == 1 && x < FLT_EPSILON / 128) { 131 *sqrt_A2my2 = sqrtf(x) * sqrtf((A + y) / 2); 132 } else if (x >= FLT_EPSILON * fabsf(y - 1)) { 133 Amy = f(x, y + 1, R) + f(x, y - 1, S); 134 *sqrt_A2my2 = sqrtf(Amy * (A + y)); 135 } else if (y > 1) { 136 *sqrt_A2my2 = x * (4 / FLT_EPSILON / FLT_EPSILON) * y / 137 sqrtf((y + 1) * (y - 1)); 138 *new_y = y * (4 / FLT_EPSILON / FLT_EPSILON); 139 } else { 140 *sqrt_A2my2 = sqrtf((1 - y) * (1 + y)); 141 } 142 } 143 } 144 145 float complex 146 casinhf(float complex z) 147 { 148 float x, y, ax, ay, rx, ry, B, sqrt_A2my2, new_y; 149 int B_is_usable; 150 float complex w; 151 152 x = crealf(z); 153 y = cimagf(z); 154 ax = fabsf(x); 155 ay = fabsf(y); 156 157 if (isnan(x) || isnan(y)) { 158 if (isinf(x)) 159 return (cpackf(x, y + y)); 160 if (isinf(y)) 161 return (cpackf(y, x + x)); 162 if (y == 0) 163 return (cpackf(x + x, y)); 164 return (cpackf(x + 0.0L + (y + 0), x + 0.0L + (y + 0))); 165 } 166 167 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) { 168 if (signbit(x) == 0) 169 w = clog_for_large_values(z) + m_ln2; 170 else 171 w = clog_for_large_values(-z) + m_ln2; 172 return (cpackf(copysignf(crealf(w), x), 173 copysignf(cimagf(w), y))); 174 } 175 176 if (x == 0 && y == 0) 177 return (z); 178 179 raise_inexact(); 180 181 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4) 182 return (z); 183 184 do_hard_work(ax, ay, &rx, &B_is_usable, &B, &sqrt_A2my2, &new_y); 185 if (B_is_usable) 186 ry = asinf(B); 187 else 188 ry = atan2f(new_y, sqrt_A2my2); 189 return (cpackf(copysignf(rx, x), copysignf(ry, y))); 190 } 191 192 float complex 193 casinf(float complex z) 194 { 195 float complex w = casinhf(cpackf(cimagf(z), crealf(z))); 196 197 return (cpackf(cimagf(w), crealf(w))); 198 } 199 200 float complex 201 cacosf(float complex z) 202 { 203 float x, y, ax, ay, rx, ry, B, sqrt_A2mx2, new_x; 204 int sx, sy; 205 int B_is_usable; 206 float complex w; 207 208 x = crealf(z); 209 y = cimagf(z); 210 sx = signbit(x); 211 sy = signbit(y); 212 ax = fabsf(x); 213 ay = fabsf(y); 214 215 if (isnan(x) || isnan(y)) { 216 if (isinf(x)) 217 return (cpackf(y + y, -INFINITY)); 218 if (isinf(y)) 219 return (cpackf(x + x, -y)); 220 if (x == 0) 221 return (cpackf(pio2_hi + pio2_lo, y + y)); 222 return (cpackf(x + 0.0L + (y + 0), x + 0.0L + (y + 0))); 223 } 224 225 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) { 226 w = clog_for_large_values(z); 227 rx = fabsf(cimagf(w)); 228 ry = crealf(w) + m_ln2; 229 if (sy == 0) 230 ry = -ry; 231 return (cpackf(rx, ry)); 232 } 233 234 if (x == 1 && y == 0) 235 return (cpackf(0, -y)); 236 237 raise_inexact(); 238 239 if (ax < SQRT_6_EPSILON / 4 && ay < SQRT_6_EPSILON / 4) 240 return (cpackf(pio2_hi - (x - pio2_lo), -y)); 241 242 do_hard_work(ay, ax, &ry, &B_is_usable, &B, &sqrt_A2mx2, &new_x); 243 if (B_is_usable) { 244 if (sx == 0) 245 rx = acosf(B); 246 else 247 rx = acosf(-B); 248 } else { 249 if (sx == 0) 250 rx = atan2f(sqrt_A2mx2, new_x); 251 else 252 rx = atan2f(sqrt_A2mx2, -new_x); 253 } 254 if (sy == 0) 255 ry = -ry; 256 return (cpackf(rx, ry)); 257 } 258 259 float complex 260 cacoshf(float complex z) 261 { 262 float complex w; 263 float rx, ry; 264 265 w = cacosf(z); 266 rx = crealf(w); 267 ry = cimagf(w); 268 if (isnan(rx) && isnan(ry)) 269 return (cpackf(ry, rx)); 270 if (isnan(rx)) 271 return (cpackf(fabsf(ry), rx)); 272 if (isnan(ry)) 273 return (cpackf(ry, ry)); 274 return (cpackf(fabsf(ry), copysignf(rx, cimagf(z)))); 275 } 276 277 static float complex 278 clog_for_large_values(float complex z) 279 { 280 float x, y; 281 float ax, ay, t; 282 283 x = crealf(z); 284 y = cimagf(z); 285 ax = fabsf(x); 286 ay = fabsf(y); 287 if (ax < ay) { 288 t = ax; 289 ax = ay; 290 ay = t; 291 } 292 293 if (ax > FLT_MAX / 2) 294 return (cpackf(logf(hypotf(x / m_e, y / m_e)) + 1, 295 atan2f(y, x))); 296 297 if (ax > QUARTER_SQRT_MAX || ay < SQRT_MIN) 298 return (cpackf(logf(hypotf(x, y)), atan2f(y, x))); 299 300 return (cpackf(logf(ax * ax + ay * ay) / 2, atan2f(y, x))); 301 } 302 303 static inline float 304 sum_squares(float x, float y) 305 { 306 307 if (y < SQRT_MIN) 308 return (x * x); 309 310 return (x * x + y * y); 311 } 312 313 static inline float 314 real_part_reciprocal(float x, float y) 315 { 316 float scale; 317 uint32_t hx, hy; 318 int32_t ix, iy; 319 320 GET_FLOAT_WORD(hx, x); 321 ix = hx & 0x7f800000; 322 GET_FLOAT_WORD(hy, y); 323 iy = hy & 0x7f800000; 324 #define BIAS (FLT_MAX_EXP - 1) 325 #define CUTOFF (FLT_MANT_DIG / 2 + 1) 326 if (ix - iy >= CUTOFF << 23 || isinf(x)) 327 return (1 / x); 328 if (iy - ix >= CUTOFF << 23) 329 return (x / y / y); 330 if (ix <= (BIAS + FLT_MAX_EXP / 2 - CUTOFF) << 23) 331 return (x / (x * x + y * y)); 332 SET_FLOAT_WORD(scale, 0x7f800000 - ix); 333 x *= scale; 334 y *= scale; 335 return (x / (x * x + y * y) * scale); 336 } 337 338 float complex 339 catanhf(float complex z) 340 { 341 float x, y, ax, ay, rx, ry; 342 343 x = crealf(z); 344 y = cimagf(z); 345 ax = fabsf(x); 346 ay = fabsf(y); 347 348 if (y == 0 && ax <= 1) 349 return (cpackf(atanhf(x), y)); 350 351 if (x == 0) 352 return (cpackf(x, atanf(y))); 353 354 if (isnan(x) || isnan(y)) { 355 if (isinf(x)) 356 return (cpackf(copysignf(0, x), y + y)); 357 if (isinf(y)) 358 return (cpackf(copysignf(0, x), 359 copysignf(pio2_hi + pio2_lo, y))); 360 return (cpackf(x + 0.0L + (y + 0), x + 0.0L + (y + 0))); 361 } 362 363 if (ax > RECIP_EPSILON || ay > RECIP_EPSILON) 364 return (cpackf(real_part_reciprocal(x, y), 365 copysignf(pio2_hi + pio2_lo, y))); 366 367 if (ax < SQRT_3_EPSILON / 2 && ay < SQRT_3_EPSILON / 2) { 368 raise_inexact(); 369 return (z); 370 } 371 372 if (ax == 1 && ay < FLT_EPSILON) 373 rx = (m_ln2 - logf(ay)) / 2; 374 else 375 rx = log1pf(4 * ax / sum_squares(ax - 1, ay)) / 4; 376 377 if (ax == 1) 378 ry = atan2f(2, -ay) / 2; 379 else if (ay < FLT_EPSILON) 380 ry = atan2f(2 * ay, (1 - ax) * (1 + ax)) / 2; 381 else 382 ry = atan2f(2 * ay, (1 - ax) * (1 + ax) - ay * ay) / 2; 383 384 return (cpackf(copysignf(rx, x), copysignf(ry, y))); 385 } 386 387 float complex 388 catanf(float complex z) 389 { 390 float complex w = catanhf(cpackf(cimagf(z), crealf(z))); 391 392 return (cpackf(cimagf(w), crealf(w))); 393 } 394